Show that `a_1,a_2,...,a_n,... `form an AP where `a_n` is defined as below :
(i) `a_n=3+4n`
(ii) `a_n=9-5n`.
Also find the sum of the first 15 terms in each case.

Explanation

To show that the sequences defined by \( a_n = 3 + 4n \) and \( a_n = 9 - 5n \) form an arithmetic progression (AP), we will follow these steps:

### Part (i): \( a_n = 3 + 4n \)

1. **Find the first four terms**:
- \( a_1 = 3 + 4 \times 1 = 3 + 4 = 7 \)
- \( a_2 = 3 + 4 \times 2 = 3 + 8 = 11 \)
- \( a_3 = 3 + 4 \times 3 = 3 + 12 = 15 \)
...